International Journal of Analysis and Applications ISSN 2291-8639 Volume 4, Number 2 (2014), 122-129 http://www.etamaths.com TWO STEP MODIFIED ISHIKAWA ITERATION SCHEME FOR MULTI-VALUED MAPPINGS IN CAT(0) SPACE PANKAJ KUMAR JHADE1,∗ AND A. S. SALUJA2 Abstract. The aim of this paper is to prove some strong convergence theo- rems for the modified Ishikawa iteration scheme involving quasi-nonexpansive multi-valued mappings in the framework of CAT(0) spaces. 1. Introduction Let K be a nonempty convex subset of a Banach space X = (X,‖ · ‖). The set K is called proximal if for each x ∈ X, there exists an element y ∈ K such that ‖x−y‖ = d(x,K), where d(x,K) = inf{‖x− z‖ : z ∈ K}. Let CB(K),K(K) and P(K) denote the family of nonempty closed bounded subsets, nonempty compact subsets and nonempty proximal bounded subsets of K respectively. The Hausdorff metric on CB(K) is defined by H(A,B) = max{sup x∈A d(x,B), sup y∈B d(y,A)} for A,B ∈ CB(K). A single-valued mapping T : K → K is called nonexpansive if ‖T(x) −T(y)‖ ≤ ‖x−y‖ for x,y ∈ K. A multi-valued mapping T : K → CB(K) is said to be nonexpansive if H(T(x),T(y)) ≤‖x−y‖ for all x,y ∈ K. An element p ∈ K is called a fixed point of T : K → K (respectively, T : K → CB(K)) if p = T(p) (respectively, p ∈ T(p)). The set of fixed points of T is denoted by F(T). The mapping T : K → CB(K) is called quasi-nonexpansive [27] if F(T) 6= φ and H(T(x),T(p)) ≤ ‖x − p‖ for all x ∈ K and all p ∈ F(T). It is clear that every nonexpansive multi-valued mapping T with F(T) 6= φ is quasi-nonexpansive. But there exists quasi-nonexpansive mappings that are not nonexpansive. Example 1.1. Let K = [0,∞) with the usual metric and T : K → CB(K) be defined by T(x) =   {0}, if x ≤ 1;[ x− 3 4 ,x− 1 3 ] , if x > 1 Then clearly F(T) = {0} and for any x we have H(T(x),T(0)) ≤‖x−0‖, hence T is quasi-nonexpansive. However, if x = 2, y = 1 we get H(T(x),T(y)) > |x−y| = 1 and hence not nonexpansive. 2010 Mathematics Subject Classification. Primary: 47H10; Secondary: 54H25, 54E40. Key words and phrases. Quasi-nonexpansive multi-valued maps, Modified Ishikawa iteration scheme, fixed points, CAT(0) space. c©2014 Authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the Creative Commons Attribution License. 122 TWO STEP MODIFIED ISHIKAWA ITERATION SCHEME 123 The mapping T : K → CB(K) is called hemi-compact if, for any sequence {xnk} of {xn} such that xnk → p ∈ K. We note that if K is compact, then every multi- valued mapping T : K → CB(X) is hemi-compact . T : K → CB(K) is said to satisfy Condition (I )[24], if there is a nondecreasing function f : [0,∞) → [0,∞) with f(0) = 0, f(r) > 0 for r ∈ (0,∞) such that d(x,T(x)) ≥ f(d(x,F(T))) for all x ∈ K. Iterative techniques for approximating fixed points of nonexpansive single-valued mappings have been studied by various authors (see; [24],[30],[11],[22]) using the Mann iteration scheme or the Ishikawa iteration scheme. For details on the subject, we refer the reader to Berinde [2]. Sastry and Babu [23] studied the Mann and Ishikawa iteration schemes for multi- valued mappings and proved that these schemes for a multi-valued map T with a fixed point p converges to a fixed point q of T under certain conditions. They also claimed that the fixed point q may be different from p. Panyanak [21] extended the result of Sastry and Babu [23] by modifying the iteration schemes of Sastry and Babu [23] in the setting of uniformly convex Banach spaces but the domain of T remains compact. Song and Wang [28,29] noted that there was a gap in the proof of Theorem 3.1 of [21] and Theorem 5 of [23]. Because the iteration xn depends on a fixed point p ∈ F(T) as well as T . If q ∈ F(T) and q 6= p, then the iteration xn defined by q is different from the one defined by p. Therefore, one cannot derive the monotonicity of sequence {‖xn −q‖} from the monotonicity of {‖xn −p‖}. So the conclusion of Theorem 3.1 [21] and Theorem 5 [23] are very dubious. They further solved/revised the gap and also gave the affirmative answer to the above question using the following Ishikawa iteration scheme. yn = βnzn + (1 −βn)xn, xn+1 = αnz ′ n + (1 −αn)xn where ‖zn−z ′ n‖≤ H(T(xn),T(yn))+γn and ‖zn+1−z ′ n‖≤ H(T(xn+1),T(yn))+γn for zn ∈ T(xn) and z ′ n ∈ T(yn). Recently, Shahzad and Zegeye [26] introduced the modified Ishikawa iteration schemes as follows: (SZ1): Let K be a nonempty convex subset of a Banach space X and αn,βn ∈ [0, 1]. The sequence of Ishikawa iterates is defined by x0 ∈ K, yn = βnzn + (1 −βn)xn, n ≥ 0 where zn ∈ T(xn), and xn+1 = αnz ′ n + (1 −αn)xn, n ≥ 0 where z ′ n ∈ T(yn) They also proved some interesting results on the strong convergence of the se- quence defined by (SZ1). Motivated and inspired by the above work, we introduced the following modified Ishikawa iteration schemes and prove some strong convergence theorems for these schemes in the setting of CAT(0) space. 124 JHADE AND SALUJA Modified Ishikawa Iteration Scheme: (PS1): Let K be a nonempty convex subset of a complete CAT(0) space X and αn,βn ∈ [0, 1]. The sequence of Ishikawa iterates is defined by x0 ∈ K, yn = βnzn + (1 −βn)xn, n ≥ 0 where zn ∈ T(xn), and xn+1 = αnz ′ n + (1 −αn)zn, n ≥ 0 where z ′ n ∈ T(yn) The aim of this paper, is to prove strong convergence theorems of the modified Ishikawa iteration scheme (PS1) in the setting of CAT(0) space. 2. Preliminaries A metric space X is a CAT(0) space if it is geodesically connected, and if every geodesic triangle in X is at least as ’thin’ as its comparison triangle in the Euclidean plane. It is well-known that any complete, simply connected Riemannian manifold having nonpositive sectional curvature is a CAT(0) space. Other examples include Pre-Hilbert spaces, R-trees (see [3]), Euclidean buildings (see [4]), the complex Hilbert ball with a hyperbolic metric (see [10]), and many others. For a thorough discussion of these spaces and of the fundamental role they play in geometry see Bridson and Haefliger [3]. Fixed point theory in a CAT(0) space was first studied by Kirk (see [12] and [15]). He showed that every nonexpansive (single-valued) mapping defined on a bounded closed convex subset of a complete CAT(0) space always has a fixed point. S- ince then the fixed point theory for single-valued and multi-valued mappings in CAT(0) spaces has been rapidly developed and much papers have appeared (see, e.g., [9],[17],[25],[26],[6]-[8], [12]-[13]). Let (X,d) be a metric space. A geodesic path joining x ∈ X to y ∈ X (or, more briefly, a geodesic from x to y) is a map c from a closed interval [0, l] ⊂ R to X such that c(0) = x, c(l) = y, and d(c(t),c(t′)) = |t − t′| for all t,t′ ∈ [0, l].In particular, c is an isometry and d(x,y) = l. The image α of c is called a geodesic (or metric) segment joining x and y. When it is unique this geodesic segment is denoted by [x,y]. The space (X,d) is said to be a geodesic space if every two points of X are joined by a geodesic, and X is said to be uniquely geodesic if there is exactly one geodesic joining x and y for each x,y ∈ X. A subset Y ⊆ X is said to be convex if Y includes every geodesic segment joining any two of its points. A geodesic triangle ∆(x1,x2,x3) in a geodesic metric space (X,d) consists of three points x1,x2,x3 in X (the vertices of ∆) and a geodesic segment between each pair of vertices (the edges of ∆). A comparison triangle for the geodesic triangle ∆(x1,x2,x3) in (X,d) is a triangle ∆(x1,x2,x3) := ∆(x̄1, x̄2, x̄3) in the Euclidean plane E2 such that dE2 (x̄i, x̄j) = d(xi,xj) for i,j ∈{1, 2, 3}. A geodesic space is said to be a CAT(0) space if all geodesic triangles of appropriate size satisfy the following comparison axiom. CAT(0): Let ∆ be a geodesic triangle in X and let ∆ be a comparison triangle for ∆. Then ∆ is said to satisfy the CAT(0) inequality if for all x,y ∈ ∆ and all comparison points x̄, ȳ ∈ ∆̄, d(x,y) ≤ dE2 (x̄, ȳ) TWO STEP MODIFIED ISHIKAWA ITERATION SCHEME 125 If x,y1,y2 are points in a CAT(0) space and if y0 is the midpoint of the segment [y1,y2], then the CAT(0) inequality implies (2.1) d(x,y0) 2 ≤ 1 2 d(x,y1) 2 + 1 2 d(x,y2) 2 − 1 4 d(y1,y2) 2 This is the (CN) inequality of Bruhat and Tits [5]. In fact (cf. [11], p.163), a geodesic space is a CAT(0) space if and only if it satisfies the (CN) inequality. In the sequel, we need the following lemmas which will be used frequently in the proofs of our main results. Lemma 2.1. ([21]) Let {αn},{βn} be two real sequences such that (1) 0 ≤ αn,βn < 1; (2) βn → 0 as n →∞; (3) ∑ αnβn = ∞ Let {γn} be a nonnegative real sequence such that ∑ αnβn(1 − βn)γn is bounded. Then {γn} has a subsequence which converges to zero. Lemma 2.2. ([3, Proposition 2.4]) Let (X,d) be a CAT(0) space.Let K be a subset of X which is complete in the induced metric. Then, for every x ∈ X, there exists a unique point P(x) ∈ K such that d(x,P(x)) = inf{d(x,y) : y ∈ K}. Moreover, the map x 7→ P(x) is a nonexpansive retract from X into K. Lemma 2.3. ([8, Lemma 2.1(iv)]) Let (X,d) be a CAT(0) space. For x,y ∈ X and t ∈ [0, 1], there exists a unique point z ∈ [x,y] such that (2.2) d(x,z) = td(x,y) and d(y,z) = (1 − t)d(x,y) We use the notation (1 − t)x⊕ ty for the unique point z satisfying (2.2). Lemma 2.4. ([8, Lemma 2.4]) Let Let (X,d) be a CAT(0) space. For x,y,z ∈ X and t ∈ [0, 1], we have d((1 − t)x⊕ ty,z) ≤ (1 − t)d(x,z) + td(y,z) Lemma 2.5. ([8, Lemma 2.5]) Let Let (X,d) be a CAT(0) space. For x,y,z ∈ X and t ∈ [0, 1], we have d((1 − t)x⊕ ty,z)2 ≤ (1 − t)d(x,z)2 + td(y,z)2 − t(1 − t)d(x,y)2 3. Main Results Lemma 3.1. Let K be a nonempty compact convex subset of a complete CAT(0) space X, and let T : K → CB(K) be a quasi-nonexpansive multi-valued mapping. Suppose that lim n→∞ d(xn,T(xn)) = 0 for some sequence {xn} in K. Then T has a fixed point. Moreover, if {d(xn,y)} converges for each y ∈ F(T), then {xn} strongly converges to a fixed point of T . Proof. By the compactness of K, there exists a subsequence {xnk} of {xn} such that xnk → q ∈ K. Thus d(q,Tq) ≤ d(q,xnk ) + d(xnk,Txnk ) + H(Txnk,q) → 0 as,k →∞ 126 JHADE AND SALUJA This implies that q is a fixed point of T. Since the limit of {d(xn,q)} exists and limk→∞ d(xnk,q) = 0, we have limk→∞ d(xn,q) = 0. This shows that the sequence {xn} converges strongly to a fixed point of q ∈ K. � Theorem 3.2. Let K be a nonempty compact convex subset of a complete CAT(0) space X, and let T : K → CB(K) be a quasi-nonexpansive multi-valued mapping and F(T) 6= φ satisfying T(p) = {p} for any fixed point p ∈ F(T). Let {xn} be the sequence of Ishikawa iterates defined by PS1. Assume that (1) αn,βn ∈ [0, 1); (2) limn→∞ βn = 0; (3) ∑∞ n=0 αnβn = ∞. Then the sequence {xn} strongly converges to a fixed point of T . Proof. Let p ∈ F(T). Then by Lemma 2.5, we have d(xn+1,p) 2 = d((1 −αn)zn ⊕αnz ′ n,p) 2 ≤ (1 −αn)d(zn,p)2 + αnd(z ′ n,p) 2 −αn(1 −αn)d(zn,z ′ n) 2 ≤ (1 −αn)(H(Txn,Tp))2 + αn(H(Tyn,Tp))2 −αn(1 −αn)d(zn,z ′ n) 2 ≤ (1 −αn)d(xn,p)2 + αnd(yn,p)2 −αn(1 −αn)d(zn,z ′ n) 2 ≤ (1 −αn)d(xn,p)2 + αnd(yn,p)2 Also d(yn,p) = d(βnzn ⊕ (1 −βn)xn,p)2 ≤ (1 −βn)d(xn,p)2 + βnd(zn,p)2 −βn(1 −βn)d(xn,zn)2 ≤ (1 −βn)d(xn,p)2 + βn(H(Txn,Tp))2 −βn(1 −βn)d(xn,zn)2 ≤ (1 −βn)d(xn,p)2 + βnd(xn,p)2 −βn(1 −βn)d(xn,zn)2 ≤ d(xn,p)2 −βn(1 −βn)d(xn,zn)2 So d(xn+1,p) 2 ≤ (1 −αn)d(xn,p)2 + αnd(xn,p)2 −αnβn(1 −βn)d(xn,zn)2 ≤ d(xn,p)2 −αnβn(1 −βn)d(xn,zn)2 This implies (3.1) d(xn+1,p) 2 ≤ d(xn,p)2 (3.2) αnβn(1 −βn)d(xn,zn)2 ≤ d(xn,p)2 −d(xn+1,p)2 It follows from (3.1) that the sequence {d(xn,p)} is decreasing and hence limn→∞ d(xn,p) exists. On the other hand (3.2) implies ∞∑ n=0 αnβn(1 −βnd(xn,zn)2 ≤ d(x1,p)2 ≤∞ Then by Lemma 2.1, there exists a subsequence {d(xnk,znk )} of d(xn,zn) such that lim k→∞ d(xnk,znk ) = 0 TWO STEP MODIFIED ISHIKAWA ITERATION SCHEME 127 This implies lim k→∞ d(xnk,Txnk ) = 0 By Lemma 3.1, {xnk} converges to a point q ∈ F(T). Since the limit of {d(xn,q)} exists, it must be the case that limn→∞ d(xn,q) = 0 and hence the conclusion follows. � Theorem 3.3. Let K be a nonempty compact convex subset of a complete CAT(0) space X, and let T : K → CB(K) be a quasi-nonexpansive multi-valued mapping that satisfying Condition (I). Let {xn} be the sequence of Ishikawa iterates defined by PS1. Assume that F(T) 6= φ satisfying T(p) = {p} for any fixed point p ∈ F(T) and αn,βn ∈ [a,b] ⊂ (0, 1). Then the sequence {xn} converges strongly to a fixed point of T . Proof. Similar to the proof of Theorem 3.2, we obtain limn→∞ d(xn,p) exists for all p ∈ F(T) and (3.3) αnβn(1 −βn)d(xn,zn)2 ≤ d(xn,p)2 −d(xn+1,p)2 Then (3.4) a2(1 − b)d(xn,zn)2 ≤ αnβn(1 −βn)d(xn,zn)2 ≤ d(xn,p)2 −d(xn+1,p)2 This implies (3.5) ∞∑ n=0 a2(1 − b)d(xn,zn)2 ≤ d(x1,p)2 < ∞ Thus, limn→∞ d(xn,zn) 2. Since zn ∈ T(xn), (3.6) d(xn,T(xn)) ≤ d(xn,zn) Therefore limn→∞ d(xn,T(xn)) = 0. Furthermore, (3.7) lim n→∞ d(xn,F(T)) = 0 The proof of remaining part closely follows the proof of [21, Theorem 3.8], simply replacing ‖ ·‖ with d(·, ·). � Corollary 3.4. Let K be a nonempty compact convex subset of a complete CAT(0) space X, and let T : K → CB(K) be a nonexpansive multi-valued mapping that satisfying Condition (I). Let {xn} be the sequence of Ishikawa iterates defined by PS1. Assume that F(T) 6= φ satisfying T(p) = {p} for any fixed point p ∈ F(T) and αn,βn ∈ [a,b] ⊂ (0, 1). Then the sequence {xn} converges strongly to a fixed point of T . Theorem 3.5. Let K be a nonempty compact convex subset of a complete CAT(0) space X, and let T : K → CB(K) be a quasi-nonexpansive multi-valued mapping and F(T) 6= φ satisfying T(p) = {p} for any fixed point p ∈ F(T). Let {xn} be the sequence of Ishikawa iterates defined by PS1. Assume that T is hemicompact and continuous, and (1) αn,βn ∈ [0, 1); (2) limn→∞ βn = 0; (3) ∑∞ n=0 αnβn = ∞. 128 JHADE AND SALUJA Then the sequence {xn} strongly converges to a fixed point of T . Proof. Let p ∈ F(T). Then, from 3.2 αnβn(1 −βn)d(xn,zn)2 ≤ d(xn,p)2 −d(xn+1,p)2 which implies that ∞∑ n=0 αnβn(1 −βnd(xn,zn)2 ≤ d(x1,p)2 < ∞ Thus, limn→∞ d(xn,zn) = 0. Since d(xn,T(xn)) ≤ d(xn,zn) → 0 as n →∞ and T is hemicompact, there is a subsequence {xnk} of {xn} such that xnk → q for some q ∈ K. Since T is continuous, we have d(xnk,T(xnk )) → d(q,T(q)). Thus, we have d(q,T(q)) = 0 and so q ∈ F(T). By Theorem 3.2 limn→∞ d(xn,p) exists for each p ∈ F(T), it follows that {xn} converges strongly to q. 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Appl., 178(1993), 301-308. 1Department of Mathematics, NRI Institute of Information Science & Technology, Bhopal-462021 , INDIA 2Department of Mathematics, J. H. Government (PG) College, Betul 460001, INDIA ∗Corresponding author